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Register a new userThe Resolution of Kellers Conjecture
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Added by
Joshua Brakensiek
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We consider three graphs, $G_{7,3}$, $G_{7,4}$, and $G_{7,6}$, related to Kellers conjecture in dimension 7. The conjecture is false for this dimension if and only if at least one of the graphs contains a clique of size $2^7 = 128$. We present an automated method to solve this conjecture by encoding the existence of such a clique as a propositional formula. We apply satisfiability solving combined with symmetry-breaking techniques to determine that no such clique exists. This result implies that every unit cube tiling of $mathbb{R}^7$ contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of $mathbb{R}^8$ exists (which we also verify), this completely resolves Kellers conjecture.
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The graph reconstruction conjecture asserts that every finite simple graph on at least three vertices can be reconstructed up to isomorphism from its deck - the collection of its vertex-deleted subgraphs. Kocays Lemma is an important tool in graph reconstruction. Roughly speaking, given the deck of a graph $G$ and any finite sequence of graphs, it gives a linear constraint that every reconstruction of $G$ must satisfy. Let $psi(n)$ be the number of distinct (mutually non-isomorphic) graphs on $n$ vertices, and let $d(n)$ be the number of distinct decks that can be constructed from these graphs. Then the difference $psi(n) - d(n)$ measures how many graphs cannot be reconstructed from their decks. In particular, the graph reconstruction conjecture is true for $n$-vertex graphs if and only if $psi(n) = d(n)$. We give a framework based on Kocays lemma to study this discrepancy. We prove that if $M$ is a matrix of covering numbers of graphs by sequences of graphs, then $d(n) geq mathsf{rank}_mathbb{R}(M)$. In particular, all $n$-vertex graphs are reconstructible if one such matrix has rank $psi(n)$. To complement this result, we prove that it is possible to choose a family of sequences of graphs such that the corresponding matrix $M$ of covering numbers satisfies $d(n) = mathsf{rank}_mathbb{R}(M)$.
As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order in a similar way as Devlin characterised big Ramsey degrees of the generic linear order (the order of rationals).
A $k$-linear coloring of a graph $G$ is an edge coloring of $G$ with $k$ colors so that each color class forms a linear forest -- a forest whose each connected component is a path. The linear arboricity $chi_l(G)$ of $G$ is the minimum integer $k$ such that there exists a $k$-linear coloring of $G$. Akiyama, Exoo and Harary conjectured in 1980 that for every graph $G$, $chi_l(G)leq left lceil frac{Delta(G)+1}{2}rightrceil$ where $Delta(G)$ is the maximum degree of $G$. First, we prove the conjecture for 3-degenerate graphs. This establishes the conjecture for graphs of treewidth at most 3 and provides an alternative proof for the conjecture in some classes of graphs like cubic graphs and triangle-free planar graphs for which the conjecture was already known to be true. Next, for every 2-degenerate graph $G$, we show that $chi_l(G)=leftlceilfrac{Delta(G)}{2}rightrceil$ if $Delta(G)geq 5$. We conjecture that this equality holds also when $Delta(G)in{3,4}$ and show that this is the case for some well-known subclasses of 2-degenerate graphs. All our proofs can be converted into linear time algorithms.
We prove a conjecture of Ohba which says that every graph $G$ on at most $2chi(G)+1$ vertices satisfies $chi_ell(G)=chi(G)$.
Motivated by a hat guessing problem proposed by Iwasawa cite{Iwasawa10}, Butler and Graham cite{Butler11} made the following conjecture on the existence of certain way of marking the {em coordinate lines} in $[k]^n$: there exists a way to mark one point on each {em coordinate line} in $[k]^n$, so that every point in $[k]^n$ is marked exactly $a$ or $b$ times as long as the parameters $(a,b,n,k)$ satisfies that there are non-negative integers $s$ and $t$ such that $s+t = k^n$ and $as+bt = nk^{n-1}$. In this paper we prove this conjecture for any prime number $k$. Moreover, we prove the conjecture for the case when $a=0$ for general $k$.
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